Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials

Previous Article
Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent
CPAA Home
This Issue
Next Article
Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian

May 2016, 15(3): 965-989. doi: 10.3934/cpaa.2016.15.965

Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials

Weiwei Ao ^1, , Liping Wang ^2, and Wei Yao ^3,

1.	Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2
2.	Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241
3.	Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Chile

Received October 2015 Revised November 2015 Published February 2016

Abstract
Related Papers

Without any symmetric conditions on potentials, we proved the following nonlinear Schrödinger system \begin{eqnarray} \left\{\begin{array}{ll} \Delta u-P(x)u+\mu_1u^3+\beta uv^2=0, \quad &\mbox{in} \quad R^2\\ \Delta v-Q(x)v+\mu_2v^3+\beta vu^2=0, \quad &\mbox{in} \quad R^2 \end{array} \right. \end{eqnarray} has infinitely many non-radial solutions with suitable decaying rate at infinity of potentials $P(x)$ and $Q(x)$. This is the continued work of [8]. Especially when $P(x)$ and $Q(x)$ are symmetric, this result has been proved in [18].

Keywords: intermediate reduction method., non-symmetric potentials, infinitely many solutions, Schrödinger system.

Mathematics Subject Classification: Primary: 35B35, 35J25; Secondary: 35B40, 92D2.

Citation: Weiwei Ao, Liping Wang, Wei Yao. Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials. Communications on Pure and Applied Analysis, 2016, 15 (3) : 965-989. doi: 10.3934/cpaa.2016.15.965

References:

[1]	A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris Ser., 1342 (2006), 453-458. doi: 10.1016/j.crma.2006.01.024.
[2]	T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Cal. Var. Partial Differential Equations., 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.
[3]	T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.
[4]	J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Memoirs of the American Mathematical Society, 229 (2014), 89pp.
[5]	K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations, Phys. Rev. Lett. A, 285 (2001), 319-326. doi: 10.1016/S0375-9601(01)00369-3.
[6]	M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincar Anal. Non Linaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X.
[7]	N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincar Anal. Non Linaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.
[8]	M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Car. Var. PDE., 53 (2015), 473-523. doi: 10.1007/s00526-014-0756-3.
[9]	Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., ().
[10]	F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928.
[11]	T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$, Comm. Math Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.
[12]	T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Phy. D, 220 (2006), 99-115. doi: 10.1016/j.physd.2006.07.009.
[13]	Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system, Comm. math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.
[14]	A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$, Adv. Math., 221 (2009), 1843-1909.
[15]	M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-883. doi: 10.1038/43136.
[16]	M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953. doi: 10.4171/JEMS/351.
[17]	B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.
[18]	S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0.
[19]	E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.
[20]	S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates, Arch. Rat. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y.
[21]	L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615. doi: 10.1090/S0002-9947-10-04955-X.
[22]	L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc., 102 (2011), 1099-1126. doi: 10.1112/plms/pdq051.
[23]	L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem, Discrete Contin. Dyn. Syst., 34 (2014), 2333-2357.
[24]	J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.
[25]	J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.
[26]	J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1.
[27]	J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, CPAA, 11 (2012), 1003-1011.

show all references

References:

[1]	A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris Ser., 1342 (2006), 453-458. doi: 10.1016/j.crma.2006.01.024.
[2]	T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Cal. Var. Partial Differential Equations., 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.
[3]	T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.
[4]	J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Memoirs of the American Mathematical Society, 229 (2014), 89pp.
[5]	K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations, Phys. Rev. Lett. A, 285 (2001), 319-326. doi: 10.1016/S0375-9601(01)00369-3.
[6]	M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincar Anal. Non Linaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X.
[7]	N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincar Anal. Non Linaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.
[8]	M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Car. Var. PDE., 53 (2015), 473-523. doi: 10.1007/s00526-014-0756-3.
[9]	Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., ().
[10]	F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928.
[11]	T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$, Comm. Math Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.
[12]	T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Phy. D, 220 (2006), 99-115. doi: 10.1016/j.physd.2006.07.009.
[13]	Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system, Comm. math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.
[14]	A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$, Adv. Math., 221 (2009), 1843-1909.
[15]	M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-883. doi: 10.1038/43136.
[16]	M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953. doi: 10.4171/JEMS/351.
[17]	B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.
[18]	S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0.
[19]	E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.
[20]	S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates, Arch. Rat. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y.
[21]	L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615. doi: 10.1090/S0002-9947-10-04955-X.
[22]	L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc., 102 (2011), 1099-1126. doi: 10.1112/plms/pdq051.
[23]	L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem, Discrete Contin. Dyn. Syst., 34 (2014), 2333-2357.
[24]	J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.
[25]	J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.
[26]	J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1.
[27]	J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, CPAA, 11 (2012), 1003-1011.

[1]	Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242
[2]	Weiming Liu, Chunhua Wang. Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109
[3]	Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094
[4]	Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4705-4736. doi: 10.3934/dcds.2021054
[5]	Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
[6]	Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071
[7]	Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265
[8]	Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104
[9]	Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure and Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004
[10]	Lixi Wen, Wen Zhang. Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022109
[11]	Ziheng Zhang, Rong Yuan. Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials. Communications on Pure and Applied Analysis, 2014, 13 (2) : 623-634. doi: 10.3934/cpaa.2014.13.623
[12]	Dušan D. Repovš. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 401-411. doi: 10.3934/dcdss.2019026
[13]	Veronica Felli, Alberto Ferrero, Susanna Terracini. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3895-3956. doi: 10.3934/dcds.2012.32.3895
[14]	Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431
[15]	Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[16]	Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[17]	Michael Röckner, Jiyong Shin, Gerald Trutnau. Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3219-3237. doi: 10.3934/dcdsb.2016095
[18]	Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099
[19]	Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[20]	Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Weiwei Ao Liping Wang Wei Yao
on Google Scholar
Weiwei Ao Liping Wang Wei Yao

[Back to Top]